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HYBRID DISCONTINUOUS GALERKIN METHODS WITH RELAXED H(DIV)-CONFORMITY FOR INCOMPRESSIBLE FLOWS. PART I∗

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arXiv 2017年

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a Hybrid Discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favourable properties such as pointwise divergence free solutions and pressure-robustness. However, for the approximation of the velocity with a polynomial degree k it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k−1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local post-processing) this is sub-optimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k−1 are involved for normal-continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to loose the benefits of H(div)-conformity we introduce a cheap reconstruction operator which restores pressure-robustness and pointwise divergence free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis and demonstrate the performance of the method on numerical *** Codes 35Q30, 65N12, 65N22, 65N30 Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

CUT FINITE ELEMENT DISCRETIZATIONS OF CELL-BY-CELL EMI ELECTROPHYSIOLOGY MODELS

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arXiv 2023年

作者： Berre, Nanna Rognes, Marie E. Massing, André Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway Department for Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway

The EMI (Extracellular-Membrane-Intracellular) model describes electrical activity in excitable tissue, where the extracellular and intracellular spaces and cellular membrane are explicitly represented. The model couples a system of partial differential equations in the intracellular and extracellular spaces with a system of ordinary differential equations on the membrane. A key challenge for the EMI model is the generation of high-quality meshes conforming to the complex geometries of brain cells. To overcome this challenge, we propose a novel cut finite element method (CutFEM) where the membrane geometry can be represented independently of a structured and easy-to-generated background mesh for the remaining computational domain. Starting from a Godunov splitting scheme, the EMI model is split into separate PDE and ODE parts. The resulting PDE part is a non-standard elliptic interface problem, for which we devise two different CutFEM formulations: one single-dimensional formulation with the intra/extracellular electrical potentials as unknowns, and a multi-dimensional formulation that also introduces the electrical current over the membrane as an additional unknown leading to a penalized saddle point problem. Both formulations are augmented by suitably designed ghost penalties to ensure stability and convergence properties that are insensitive to how the membrane surface mesh cuts the background mesh. For the ODE part, we introduce a new unfitted discretization to solve the membrane bound ODEs on a membrane interface that is not aligned with the background mesh. Finally, we perform extensive numerical experiments to demonstrate that CutFEM is a promising approach to efficiently simulate electrical activity in geometrically resolved brain *** Codes 65M60, 65M85, 65N30, 65N85, 92C20 Copyright © 2023, The Authors. All rights reserved.

关键词： Electrophysiology

numerical analysis of a hybridized discontinuous Galerkin method for the Cahn-Hilliard problem

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arXiv 2023年

作者： Kirk, Keegan L.A. Masri, Rami Riviere, Beatrice Department of Computational and Applied Mathematics Rice University Houston United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway

The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin *** Codes 65M15 (Primary) 65M60 (Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： numerical methods

A fast and oblivious matrix compression algorithm for Volterra integral operators

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arXiv 2021年

作者： Dölz, Jürgen Egger, Herbert Shashkov, Vsevolod Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Numerical Analysis and Scientific Computing Department of Mathematics TU Darmstadt Dolivostr. 15 Darmstadt64293 Germany

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra-integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N) active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an H-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to H2matrix compression techniques. Following this idea, we formulate a variant of the H2-matrix vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N) operations and O(log N) active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a-priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems. Copyright © 2021, The Authors. All rights reserved.

关键词： Matrix algebra

Conservation Properties of the Augmented Basis Update & Galerkin Integrator for Kinetic Problems

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SSRN

SSRN 2023年

作者： Einkemmer, Lukas Daniel Kusch, Jonas Schotthöfer, Steffen University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Scientific Computing Ås Norway Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

numerical simulations of kinetic problems can become prohibitively expensive due to their large memory footprint and computational costs. A method that has proven to successfully reduce these costs is the dynamical low-rank approximation (DLRA). One key question when using DLRA methods is the construction of robust time integrators that preserve the invariances and associated conservation laws of the original problem. In this work, we demonstrate that the augmented basis update & Galerkin integrator (BUG) preserves solution invariances and the associated conservation laws when using a conservative truncation step and an appropriate time and space discretization. We present numerical comparisons to existing conservative integrators and discuss advantages and disadvantages. © 2023, The Authors. All rights reserved.

关键词： Integral equations

A note on terminology in multigrid methods

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A note on terminology in multigrid methods

Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Birken, Philipp Bull, Jonathan Jameson, Antony Lund University Centre for the Mathematical Sciences Numerical Analysis Box 118 Lund22100 Sweden Uppsala University Division of Scientific Computing Dept. of Information Technology Box 337 Uppsala75105 Sweden Stanford University Department of Aeronautics & Astronautics StanfordCA94305 United States

We compare terminology used in the literature on multigrid methods for compressible computational fluid dynamics to that used in linear multigrid theory. Several popular iterative and direct smoothers are presented side-by-side using the same terminology. We argue for greater analysis of these methods in order to place them into a more rigorous framework and to identify the most promising candidates for future development. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Terminology

Conservation properties of the augmented basis update & Galerkin integrator for kinetic problems

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arXiv 2023年

作者： Einkemmer, Lukas Kusch, Jonas Schotthöfer, Steffen University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Scientific Computing Ås Norway Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

关键词： Kinetics

Modeling stochastic eye tracking data: A comparison of quantum generative adversarial networks and Markov models

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arXiv 2024年

作者： Bhandari, Shailendra Lencastre, Pedro Lind, Pedro Department of Computer Science OsloMet – Oslo Metropolitan University OsloN-0130 Norway Simula Research Laboratory Numerical Analysis and Scientific Computing Oslo0164 Norway

We explore the use of quantum generative adversarial networks QGANs for modeling eye movement velocity data. We assess whether the advanced computational capabilities of QGANs can enhance the modeling of complex stochastic distribution beyond the traditional mathematical models, particularly the Markov model. The findings indicate that while QGANs demonstrate potential in approximating complex distributions, the Markov model consistently outperforms in accurately replicating the real data distribution. This comparison underlines the challenges and avenues for refinement in time series data generation using quantum computing techniques. It emphasizes the need for further optimization of quantum models to better align with real-world data characteristics. © 2024, CC BY.

关键词： Generative adversarial networks

Risk-averse optimal control of random elliptic variational inequalities

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arXiv 2022年

作者： Alphonse, Amal Geiersbach, Caroline Hintermüller, Michael Surowiec, Thomas M. Weierstrass Institute Mohrenstrasse 39 Berlin10117 Germany Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems